Polar phase of superfluid 3 He: Dirac lines in the parameter and momentum spaces
PPolar phase of superfluid He: Dirac lines in the parameter and momentum spaces
G.E. Volovik
1, 2 Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland Landau Institute for Theoretical Physics, acad. Semyonov av., 1a, 142432, Chernogolovka, Russia (Dated: January 17, 2018)The time reversal symmetric polar phase of the spin-triplet superfluid He has two types of Diracnodal lines. In addition to the Dirac loop in the spectrum of the fermionic Bogoliubov quasiparticlesin the momentum space ( p x , p y , p z ), the spectrum of bosons (magnons) has Dirac loop in the 3Dspace of parameters – the components of magnetic field ( H x , H y , H z ). The bosonic Dirac systemlives on the border between the type-I and type-II. PACS numbers:
Originally the topology of the points and lines of levelcrossing (diabolical points ) has been investigatedin a parameter space. In particular, while encirclinga diabolical point in the space of two parameters, thewavefunction changes sign.
Typically this has been ap-plied to electronic spectrum in molecular systems. Laterthe topological methods have been applied to the di-abolical points in the spectrum of fermionic quasipar-ticles (Bogoliubov quasipartilces) in gapless superfluidsand superconductors, where the parameter space is thespace of linear momentum in superfluids and quasimo-mentum in superconductors, or the extended phase space( p , r ). In particular, the topologically protected dia-bolical point in 3D momentum space – the Weyl point –gives rise to Weyl fermions and effective gauge and grav-ity fields emerging in the vicinity of the Weyl point.
This analog of relativistic quantum field allowed to ex-perimentally verify the Adler-Bell-Jackiw equationfor chiral anomaly in chiral superfluid He-A. Then thistopological consideration has been extended to the spec-trum of bosonic excitations, see e.g. .Recently the new trend is towards the topology in theextended space, which combines the momentum spaceand the parameter space, see e.g. Here we show thatthe appropriate system, where the two spaces (momen-tum space and parameter space) are topologically con-nected, is the polar phase of superfluid He discovered innematically ordered aerogel. In momentum space the polar phase contains the Diracnodal line in the quasiparticle spectrum determined bythe 2 × H ( p ) = v F ( p − p F ) τ + ∆ P ˆ m · ˆ p τ . (1)Here τ a are the Pauli matrices in the Bogoliubov-Nambuspace; p F and v F are the Fermi momentum and Fermivelocity in the normal state of liquid He; ∆ P is the gapamplitude in the polar phase; ˆ p = p /p ; ˆ m is the unitvector of uniaxial anisotropy axis provided by the direc-tion of the aerogel strands, and we choose the coordinatesystems with ˆ z = ˆ m ; we ignore here the spin structureof the order parameter (but later it will be important forthe consideration of spin dynamics).The nodal line, where the spectrum of negative energystates touches the spectrum of positive energy states, is ΔΦ = π H x γ H = Ω P H z C H y ΔΦ = π p x p = p F p z m C p y FIG. 1: (Color online) Exceptional lines of level crossinganalyzed by von Neumann and Wigner in the polar phaseof superfluid He. The geometric Berry phase around theselines changes by π . Left : Dirac line in the quasiparticle spectrum in space of thecomponents of momentum ( p x , p y , p z ). At this topologicallyprotected line ( p z = 0, p = p F ) the energy of the Bogoliubovquasiparticles in Eq.(1) is zero. Right : Dirac line in the space of parameters – components ofmagnetic field ( H x , H y , H z ), which determine the frequency ofmagnons in Eqs. (3) and (4). At this topologically protectedline ( H z = 0, γH = Ω P , where Ω P is the Leggett frequency)the branch of optical magnon and the branch of light Higgsmode cross each other, see Fig. 2. at p z = 0 and p = p F , see Fig.1 ( left ). In the vicinity ofthe Dirac line there emerges the peculiar type of quan-tum electrodynamics with the non-analytic action for theeffective electromagnetic field, ( B − E ) / . Here we show that the spectrum of spin waves(magnons) – the Goldstone modes of the polar phase –also experiences the topologically protected Dirac nodalline, but now in the parameter space, see Fig.1 ( right ).This spectrum at different magnitudes and orientationsof magnetic field has been measured in Ref. . The equa-tion for magnetization M in the spin-wave modes followsfrom the Leggett equation, obtained using the free energyfor spin dynamics, see Ref. : F = 12 χ − ab M a M b − M · H + χ ⊥ γ Ω P (ˆ d · ˆ m ) . (2)Here H is the external magnetic field; γ is gyromagneticratio; the unit vector ˆ d is the spin part of the order pa- a r X i v : . [ c ond - m a t . o t h e r] J a n γ H Ω P ω light Higgs Diracpointopticalmagnon FIG. 2: For magnetic field H ⊥ ˆ m , two banches of magnonspectrum (light Higgs mode ω = Ω P and optical magnon ω = γH ) do not interact with each other and cross each other atthe exceptional point γH = Ω P . In the 3D space of magneticfield this Dirac point becomes the Dirac deneneracy line inFig.1 ( right ). These two branches form the Dirac cone, whichis on the border between the tilted and overtilted cones. Inother words, the Hamiltonian (4) describes the bosonic Diracsystem, which is on the border between the type-I and type-II. rameter, which determines the easy axis of spontaneousanisotropy of spin susceptibility χ ab . The last term inEq.(2) is the spin-orbit coupling, where Ω P is the so-called Leggett frequency, the frequency of the longitudi-nal NMR. The equation for magnetization has the fol-lowing matrix form: ω Ψ = H ( H )Ψ , (3) H ( H ) = ( γH ) + Ω P (cid:18) ( γH ) − Ω P P cos λ (cid:19) τ − Ω P sin λ cos λ τ . (4)Here the two-component function is Ψ = ( M ⊥ , M (cid:107) − M ),where M ⊥ and M (cid:107) are the transverse and longitudinalcomponents of magnetization with respect to the direc-tion of magnetic field, and M = χ ⊥ H is an equilibriummagnetization; τ a are the Pauli matrices connecting thetwo components of magnetization; ω L = γH is Larmorfrequency; λ is the angle of magnetic field with respectto anisotropy axis ˆ m , i.e. cos λ = ˆ m · H /H .For λ = π/
2, the two branches do not interact witheach other and may cross each other, see Fig. 2. In themode with ω = γH , the transverse component M ⊥ os-cillates. This mode is excited in transverse NMR exper-iments. The mode with ω = Ω P and with oscillating M (cid:107) is excited in longitudinal NMR experiments. In the otherlanguage these two branches correspond respectively tothe optical magnon and the light Higgs mode. The modes do not interact with each other only at λ = π/ λ = 0. Otherwise, these modes interact producingthe observed parametric decay of Bose-Einstein conden-sate of optical magnons to light Higgs modes, and therepulsion of the levels – the observed avoiding crossing. .At λ = π/ γH = Ω P these two branches crosseach other. This is the degeneracy point of the levelcrossing – the Dirac diabolical point in the space of thetwo parameters, γH = Ω P and λ = π/
2. If one takesinto account all three components of magnetic field H ,one obtains the Dirac line (circle) H z = 0, γH = Ω P in the 3D space of magnetic field ( H x , H y , H z ) in Fig.1( right ), where the spectrum is degenerate. Close to theDirac line, the Hamiltonian in Eq.(4) transforms to: H ( H ) − Ω P ≈ Ω P ( γH − Ω P )+Ω P ( γH − Ω P ) τ − Ω P ˆ m · ˆ h τ , (5)where ˆ h = H /H . Equation (5) is analogous to Eq.(1),with γ Ω P and Ω P /γ playing the roles of Fermi velocityand Fermi momentum, and Ω P being the analog of gapamplitude. Since the analog of the Fermi velocity coin-cides with the derivative of the first term in the right-hand side with respect to H , the Hamiltonians (4) and(5) describe the bosonic Dirac system, which is on theborder between the type-I and type-II. The Higgsmode in Fig. 2 is ”dispersionless”, dω/dH = 0, which ison the border between the tilted ( dω/dH <
0) and theovertilted ( dω/dH >
0) Dirac cones.In both cases of fermionic and bosonic spectrum in Fig.1, the Dirac nodal line has nontrivial topological charge N = 1, see e.g. N = 14 πi Tr (cid:73) C dl τ ˜ H − ∂ l ˜ H . (6)Here ˜ H is the traceless part of the matrix H , and theintegral is along the loop C in momentum or parameterspace enclosing the Dirac line. The nontrivial topologymeans that when the momentum p in Fig.1 ( left ) or mag-netic field H in Fig.1 ( right ) adiabatically evolves alongthis loop, the corresponding geometric Berry phase Φchanges by π .In conclusion, there are two topologically protectedDirac lines in the polar phase of superfluid He. Oneof them is fermionic, which lives in the 3D momen-tum space ( p x , p y , p z ). It gives rise to the peculiartype of the effective quantum electrodynamics. Theother one is bosonic and lives in the 3D parameterspace ( H x , H y , H z ). The NMR spectrum near thisDirac line has been experimentally studied in Ref. 23.The next task should be to combine the effects of thetwo Dirac lines, which form the 2D degeneracy man-ifold in the extended 6D momentum+parameter space( p x , p y , p z , H x , H y , H z ). This will involve the effects re-lated to dynamics of Bogoliubov quasiparticles near thefermionic Dirac line interacting with the spin waves invicinity of the bosonic Dirac line, such as adiabatic Thou-less pumping. I thank Tero Heikkil¨a for interesting discussions, whichresulted in this paper. This work has been supported bythe European Research Council (ERC) under the Euro- pean Union’s Horizon 2020 research and innovation pro-gramme (Grant Agreement No. 694248). M. Born and R. Oppenheimer, Zur Quantentheorie derMolekeln, Annalen der Physik , 457–484 (1927). J. von Neumann und E.P. Wigner, ¨Uber das Verhalten vonEigenwerten bei adiabatischen Prozessen, Phys. Zeit. ,467–470 (1929). M.V. Berry, Quantizing a classically ergodic system:Sinai’s billiard and the KKR method, Ann. Phys. ,163–216 (1981). M.V. Berry, Quantal phase factors accompanying adiabaticchanges, Proceedings of the Royal Society of London, Se-ries A , 45–57 (1984). M.V. Berry, Geometric phase memories, Nature Physics ,148–150 (2010). G.E. Volovik, Zeroes in the fermion spectrum in super-fluid systems as diabolical points, Pisma ZhETF , 81–84(1987); JETP Lett. , 98–102 (1987). M.M. Salomaa, G E. Volovik, Cosmiclike domain walls insuperfluid He-B: Instantons and diabolical points in ( k , r ) space, Phys. Rev. B , 9298–9311 (1988). P.G. Grinevich, G. E. Volovik, Topology of gap nodes in su-perfluid He: π homotopy group for He-B disclination,”J. Low Temp. Phys. , 371–380 (1988). Shinsei Ryu, A.P. Schnyder, A. Furusaki and A.W.W Lud-wig, Topological insulators and superconductors: tenfoldway and dimensional hierarchy, New J. Phys. , 065010(2010). C.D. Froggatt and H.B. Nielsen,
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